Charles Henry czhenry at gmail.com
Tue Jul 8 19:33:57 CEST 2008

```Hey, Cyrille,

I kind of thought so...  we are quickly running into the law of
diminishing returns.  I was up late, last night, working on the
analysis some more.  I think I can have another 6-point version with
better characteristics tonight.

Chuck

On Tue, Jul 8, 2008 at 11:35 AM, cyrille henry
<cyrille.henry at la-kitchen.fr> wrote:
> hello Chuck,
>
> i tested this. (and commited)
> i think tabread6c~ is a bit better than tabread4c~. but differences are more
> smaller
>
> thx
>
> Cyrille
>
>
> Charles Henry a écrit :
>>
>> On Sat, Jun 28, 2008 at 6:43 AM, cyrille henry
>> <cyrille.henry at la-kitchen.fr> wrote:
>>
>>> ok, i'll try that.
>>> but i don't think adjusting the 2nd derivative is the best thing to do.
>>> for me, having a 6 point interpolation would be more important.
>>
>> I put together a 6-point interpolation formula and analyzed it.  For
>> this I used a 5th degree polynomial, and 6 constraints:
>>
>> (I want to change up the notation a bit, and not use the letters a, b,
>> c, etc... when switching to 6-point.  Y[-2],Y[-1],Y[0], Y[1], Y[2],
>> Y[3] are the points from the table. a5 is the coefficient of x^5, a4
>> is the coeff. of x^4, ... a0 is a constant term.  f(x) is the
>> interpolation polynomial.)
>>
>> f(0)=Y[0]
>> f(1)=Y[1]
>> f'(0)= 1/12*Y[-2] - 2/3*Y[-1] + 2/3*Y[1] - 1/12*Y[2]
>> f'(1)= 1/12*Y[-1] - 2/3*Y[0]  + 2/3*Y[2] - 1/12*Y[3]
>> f''(0)= -1/12*Y[-2] + 4/3*Y[-1] - 5/2*Y[0] + 4/3*Y[1] - 1/12*Y[2]
>> f''(1)= -1/12*Y[-1] + 4/3*Y[0]  - 5/2*Y[1] + 4/3*Y[2] - 1/12*Y[3]
>>
>> This uses improved approximations for the derivative.  One advantage
>> of going to 6-point interpolation is to get better numerical
>> derivatives.  These approximations of the 1st and 2nd derivatives are
>> accurate up to a higher frequency than before.  We can also continue
>> to increase the number of points arbitrarily, without necessarily
>> having to increase the degree of the polynomial.  The degree of the
>> polynomial is only determined by the number of constraints, not the
>> number of points.
>>
>> The coefficients used in this scheme are
>>
>> a0= Y[0]
>> a1= 1/12*Y[-2] - 2/3*Y[-1] + 2/3*Y[1] - 1/12*Y[2]
>> a2= -1/24*Y[-2] + 2/3*Y[-1] - 5/4*Y[0] + 2/3*Y[1] - 1/24*Y[2]
>> a3= -3/8*Y[-2] + 13/8*Y[-1] - 35/12*Y[0] + 11/4*Y[1] - 11/8*Y[2] +
>> 7/24*Y[3]
>> a4= 13/24*Y[-2] - 8/3*Y[-1] + 21/4*Y[0] - 31/6*Y[1] + 61/24*Y[2] -
>> 1/2*Y[3]
>> a5= -5/24*Y[-2] + 25/24*y[-1] - 25/12*Y[0] + 25/12*Y[1] - 25/24*Y[2] +
>> 5/24*Y[3]
>>
>>
>> After that, I continued with the impulse response calculations and
>> spectral response calculations, which are a bit disappointing.  I'll
>> spare you the equations (for now) and post the graphs.  The new traces
>> for the 6-point interpolator are shown in green.  It's a little bit
>> hard to see, but the things to look for are the rate at which the
>> graph falls off and the locations of the peaks.  The 6-point function
>> has a flatter spectrum, which comes up closer to the Nyquist
>> frequency, and falls off faster.  These are the key characteristics of
>> the spectrum we want.  The green trace falls off according to 1/w^4,
>>
>> You can see the impulse response in the first graph along with the
>> spectrum.  The log vs. dB scale is used same as before, and secondly,
>> I've posted a linear graph, so you can see the difference between
>> functions near the Nyquist frequency (x=pi).
>>
>> It gives me some ideas for another 6-point scheme, more like
>> tabread4c~, which will fall off at a rate of 1/w^5 and have more
>> notches in the frequency response.  I'll work on it a bit, and see how
>> it goes.
>>
>> Chuck
>>
>>
>> ------------------------------------------------------------------------
>>
>>
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>>
>>
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